Many common sheet materials, ranging from natural materials such as wood to modern composites, possess approximately orthotropic symmetry. Within the approximations of thin-plate bending theory, the linear vibrational properties of such sheets are governed by four elastic constants and four damping constants (at any given frequency). A simple procedure is presented whereby all four elastic constants may be determined, quickly and with reasonable accuracy, from measurements of the resonant frequencies of low-frequency modes of thin rectangular plates with free edges. Also, at least three of the four damping constants may be determined by measuring the damping factors of the same modes-it turns out that the fourth damping constant does not usually have sufficient influence on the low-frequency modes for a reliable value to be found by this approach. The procedure is illustrated with measurements on a range of different sheet materials: wooden plates cut at different angles from the solid timber, plywood, and two very different fibre-reinforced composites. The discussion of these experimental results suggests that this simple procedure could form a valuable part of any programme of quality control, material selection or non-destructive testing involving orthotropic sheet materials.